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Properties

Label 12-1080e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.587\times 10^{18}$
Sign $1$
Analytic cond. $411344.$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 5·7-s − 2·11-s − 4·17-s − 8·19-s − 7·23-s + 3·25-s − 7·29-s + 16·31-s + 15·35-s + 4·37-s − 41-s − 2·43-s − 13·47-s + 18·49-s + 20·53-s − 6·55-s − 6·59-s + 11·61-s − 67-s + 28·71-s + 32·73-s − 10·77-s + 6·79-s − 21·83-s − 12·85-s + 66·89-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.88·7-s − 0.603·11-s − 0.970·17-s − 1.83·19-s − 1.45·23-s + 3/5·25-s − 1.29·29-s + 2.87·31-s + 2.53·35-s + 0.657·37-s − 0.156·41-s − 0.304·43-s − 1.89·47-s + 18/7·49-s + 2.74·53-s − 0.809·55-s − 0.781·59-s + 1.40·61-s − 0.122·67-s + 3.32·71-s + 3.74·73-s − 1.13·77-s + 0.675·79-s − 2.30·83-s − 1.30·85-s + 6.99·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{18} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(411344.\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{18} \cdot 5^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.144122338\)
\(L(\frac12)\) \(\approx\) \(9.144122338\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
good7 \( 1 - 5 T + p T^{2} + 32 T^{3} - 107 T^{4} - 83 T^{5} + 914 T^{6} - 83 p T^{7} - 107 p^{2} T^{8} + 32 p^{3} T^{9} + p^{5} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 21 T^{2} - 14 T^{3} + 26 p T^{4} - 58 T^{5} - 3673 T^{6} - 58 p T^{7} + 26 p^{3} T^{8} - 14 p^{3} T^{9} - 21 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 15 T^{2} + 72 T^{3} + 30 T^{4} - 540 T^{5} + 2765 T^{6} - 540 p T^{7} + 30 p^{2} T^{8} + 72 p^{3} T^{9} - 15 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + 2 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 4 T + 53 T^{2} + 148 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 7 T + 31 T^{2} + 88 T^{3} - 223 T^{4} - 1751 T^{5} - 8678 T^{6} - 1751 p T^{7} - 223 p^{2} T^{8} + 88 p^{3} T^{9} + 31 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T - 33 T^{2} - 184 T^{3} + 1693 T^{4} + 3673 T^{5} - 43786 T^{6} + 3673 p T^{7} + 1693 p^{2} T^{8} - 184 p^{3} T^{9} - 33 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 16 T + 87 T^{2} - 504 T^{3} + 5722 T^{4} - 33628 T^{5} + 138175 T^{6} - 33628 p T^{7} + 5722 p^{2} T^{8} - 504 p^{3} T^{9} + 87 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 2 T + 103 T^{2} - 136 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + T - 113 T^{2} - 56 T^{3} + 8237 T^{4} + 2023 T^{5} - 389450 T^{6} + 2023 p T^{7} + 8237 p^{2} T^{8} - 56 p^{3} T^{9} - 113 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 2 T - 89 T^{2} + 58 T^{3} + 4522 T^{4} - 6346 T^{5} - 216193 T^{6} - 6346 p T^{7} + 4522 p^{2} T^{8} + 58 p^{3} T^{9} - 89 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 13 T + 113 T^{2} + 530 T^{3} - 783 T^{4} - 34171 T^{5} - 303266 T^{6} - 34171 p T^{7} - 783 p^{2} T^{8} + 530 p^{3} T^{9} + 113 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 10 T + 171 T^{2} - 1036 T^{3} + 171 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 6 T - 81 T^{2} - 858 T^{3} + 2106 T^{4} + 29310 T^{5} + 86191 T^{6} + 29310 p T^{7} + 2106 p^{2} T^{8} - 858 p^{3} T^{9} - 81 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 11 T - 21 T^{2} + 84 T^{3} + 2449 T^{4} + 32767 T^{5} - 548954 T^{6} + 32767 p T^{7} + 2449 p^{2} T^{8} + 84 p^{3} T^{9} - 21 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + T - p T^{2} - 426 T^{3} - 179 T^{4} + 12173 T^{5} + 348238 T^{6} + 12173 p T^{7} - 179 p^{2} T^{8} - 426 p^{3} T^{9} - p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 14 T + 193 T^{2} - 1952 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 16 T + 3 p T^{2} - 1952 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 6 T - 117 T^{2} + 1186 T^{3} + 5010 T^{4} - 53358 T^{5} + 18795 T^{6} - 53358 p T^{7} + 5010 p^{2} T^{8} + 1186 p^{3} T^{9} - 117 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 21 T + 63 T^{2} + 480 T^{3} + 33201 T^{4} + 236523 T^{5} + 151594 T^{6} + 236523 p T^{7} + 33201 p^{2} T^{8} + 480 p^{3} T^{9} + 63 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 11 T + p T^{2} )^{6} \)
97 \( 1 + 30 T + 453 T^{2} + 3962 T^{3} + 14730 T^{4} - 207642 T^{5} - 3708051 T^{6} - 207642 p T^{7} + 14730 p^{2} T^{8} + 3962 p^{3} T^{9} + 453 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.23631746783155310263870722125, −4.89400189349183999247086697820, −4.86305918639765177060781547467, −4.85793484892526791064515441928, −4.75068145226081786364176912010, −4.37106135527871424871823460186, −4.22021658415090805284065796084, −4.20510548396639863344744533805, −4.05206783730449052800523374071, −3.63198678721570307674621317553, −3.50616422195252876109324100465, −3.44676341957964872566176052877, −3.40183622660197466231822835483, −2.75343088220371436054271448996, −2.66980443326249387141158824116, −2.41368456536287225806077157779, −2.31805662015664601990052223648, −2.09930460299801887100033127262, −2.03347713987646641953337287461, −1.89716432056364498896232321471, −1.78597488404390347379096335350, −1.26395374682101530142834506442, −0.916088527216555105526317289978, −0.67637749501245849817071905456, −0.48404926466239411490709372474, 0.48404926466239411490709372474, 0.67637749501245849817071905456, 0.916088527216555105526317289978, 1.26395374682101530142834506442, 1.78597488404390347379096335350, 1.89716432056364498896232321471, 2.03347713987646641953337287461, 2.09930460299801887100033127262, 2.31805662015664601990052223648, 2.41368456536287225806077157779, 2.66980443326249387141158824116, 2.75343088220371436054271448996, 3.40183622660197466231822835483, 3.44676341957964872566176052877, 3.50616422195252876109324100465, 3.63198678721570307674621317553, 4.05206783730449052800523374071, 4.20510548396639863344744533805, 4.22021658415090805284065796084, 4.37106135527871424871823460186, 4.75068145226081786364176912010, 4.85793484892526791064515441928, 4.86305918639765177060781547467, 4.89400189349183999247086697820, 5.23631746783155310263870722125

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.